On graphs which can be drawn on an orientable surface with small number of intersections on an edge

Let $k$ and $g$ be nonnegative integers. We call a graph $k$-nearly $g$-spherical, if it can be drawn on an orientable surface of genus $g$ such that each edge intersects at most $k$ other edges in inner points. It is proved that for $k\leq4$ the number of edges of a $k$-nearly $g$-spherical graph on $v$ vertices does not exceed $(k+3)(v+2g-2)$. It is also proved that the chromatic number of a $k$-nearly $g$-spherical graph does not exceed $\frac{2k+7+\sqrt{4k^2+12k+1+16(k+3)g}}2$.